Among all 256 elementary cellular automata rules, Rule 90 holds a celebrated place in geometry and number theory.

In binary notation: $$90 = 01011010_2$$ Look closely at the transition table for Rule 90: $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Pattern } (L, C, R) & 111 & 110 & 101 & 100 & 011 & 010 & 001 & 000 \\ \hline \text{Output} & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ \hline \end{array}$$ Notice that the output does not depend on the center cell $C$ at all! The new cell is $1$ if either the left cell is $1$ OR the right cell is $1$, but NOT both.

Mathematically, this is the Exclusive OR (XOR) operation, which is identical to addition modulo 2: $$C_{\text{new}} = (L + R) \pmod 2 = L \oplus R$$

The Connection to Pascal's Triangle

In Pascal's Triangle, every number is the sum of the two numbers above it: $$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$$ If you color all the odd numbers black ($1$) and all the even numbers white ($0$) — which is taking Pascal's triangle modulo 2 — you get the exact mathematical formula of Rule 90!

Starting from a single seed, Rule 90 generates the famous Sierpinski Triangle fractal:

• A self-similar geometric fractal with holes at every scale.
• Its perimeter is infinitely long, but its area is zero!
• Its fractional Hausdorff dimension is $D = \frac{\log 3}{\log 2} \approx 1.585$.

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• Fractal Self-Similarity: The large triangle is composed of 3 smaller copies of itself, each of which is composed of 3 even smaller copies, repeating infinitely!
• Linearity: Because Rule 90 uses modular addition ($L \oplus R$), it satisfies the principle of superposition: the pattern formed by multiple starting seeds is simply the XOR sum of the individual triangles!